For cheap a Nikon Bushmaster II 4-12x40.
Getting to 16x on the cheap is a bit more of a reach. You could go with a fixed SWFA SS 16x42...which might not be good for close up rodent shots, or even a Vortex Diamondback Tactical 4-16x44 for around $320 when on sale. But both of these might be overkill for a lightweight rimfire rifle.
A Nikon Prostaff 5 3.5-14x42 is in between and in your budget as well.
These are all decent quality scopes, especially for the $$$, but you could dip into that low end rung of BSA/Barska/Tasco and get crazy magnification for next to nothing...although I would bet the view is like looking through milk.
Good luck in finding something that works for you...
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